\(\int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx\) [1302]

Optimal result

Integrand size = 27, antiderivative size = 136 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}+\frac {2 b^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}{21 a^2 x^3}-\frac {2 b^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{21 a^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

-1/7*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^7+2/21*b^2*(-b*x^2+a)^(1/2)*(b*x^2 
+a)^(1/2)/a^2/x^3-2/21*b^(7/2)*(1-b^2*x^4/a^2)^(1/2)*EllipticF(b^(1/2)*x/a 
^(1/2),I)/a^(3/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {7}{4},-\frac {1}{2};-\frac {3}{4};\frac {b^2 x^4}{a^2}\right )}{7 x^7 \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

Integrate[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^8,x]
 

Output:

-1/7*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*HypergeometricPFQ[{-7/4, -1/2}, {-3/ 
4}, (b^2*x^4)/a^2])/(x^7*Sqrt[1 - (b^2*x^4)/a^2])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {339, 343, 289, 765, 762}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx\)

\(\Big \downarrow \) 339

\(\displaystyle -\frac {2}{7} b^2 \int \frac {1}{x^4 \sqrt {a-b x^2} \sqrt {b x^2+a}}dx-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}\)

\(\Big \downarrow \) 343

\(\displaystyle -\frac {2}{7} b^2 \left (\frac {b^2 \int \frac {1}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx}{3 a^2}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^3}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}\)

\(\Big \downarrow \) 289

\(\displaystyle -\frac {2}{7} b^2 \left (\frac {b^2 \sqrt {a^2-b^2 x^4} \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{3 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^3}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}\)

\(\Big \downarrow \) 765

\(\displaystyle -\frac {2}{7} b^2 \left (\frac {b^2 \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{3 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^3}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {2}{7} b^2 \left (\frac {b^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{3 a^{3/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{3 a^2 x^3}\right )-\frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{7 x^7}\)

Input:

Int[(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^8,x]
 

Output:

-1/7*(Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/x^7 - (2*b^2*(-1/3*(Sqrt[a - b*x^2] 
*Sqrt[a + b*x^2])/(a^2*x^3) + (b^(3/2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticF[A 
rcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(3*a^(3/2)*Sqrt[a - b*x^2]*Sqrt[a + b*x^2 
])))/7
 

Defintions of rubi rules used

rule 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 339

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^p*((c + d*x^2)^p/(e*(m + 1))) 
, x] - Simp[4*b*d*(p/(e^4*(m + 1)))   Int[(e*x)^(m + 4)*(a + b*x^2)^(p - 1) 
*(c + d*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c + a 
*d, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 343

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(p + 1)/ 
(a*c*e*(m + 1))), x] - Simp[b*d*((m + 4*p + 5)/(a*c*e^4*(m + 1)))   Int[(e* 
x)^(m + 4)*(a + b*x^2)^p*(c + d*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, 
 x] && EqQ[b*c + a*d, 0] && LtQ[m, -1]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (2 b^{4} \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right ) x^{7}+2 \sqrt {\frac {b}{a}}\, b^{4} x^{8}-5 \sqrt {\frac {b}{a}}\, a^{2} b^{2} x^{4}+3 \sqrt {\frac {b}{a}}\, a^{4}\right )}{21 \left (-b^{2} x^{4}+a^{2}\right ) x^{7} a^{2} \sqrt {\frac {b}{a}}}\)	\(142\)
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-2 b^{2} x^{4}+3 a^{2}\right )}{21 x^{7} a^{2}}-\frac {2 b^{4} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{21 a^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(146\)
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {\sqrt {-b^{2} x^{4}+a^{2}}}{7 x^{7}}+\frac {2 b^{2} \sqrt {-b^{2} x^{4}+a^{2}}}{21 a^{2} x^{3}}-\frac {2 b^{4} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )}{21 a^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(146\)

Input:

int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/21*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)*(2*b^4*((-b*x^2+a)/a)^(1/2)*((b*x^2 
+a)/a)^(1/2)*EllipticF(x*(b/a)^(1/2),I)*x^7+2*(b/a)^(1/2)*b^4*x^8-5*(b/a)^ 
(1/2)*a^2*b^2*x^4+3*(b/a)^(1/2)*a^4)/(-b^2*x^4+a^2)/x^7/a^2/(b/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=-\frac {2 \, b^{3} x^{7} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-1) - {\left (2 \, b^{2} x^{4} - 3 \, a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{21 \, a^{2} x^{7}} \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8,x, algorithm="fricas")
 

Output:

-1/21*(2*b^3*x^7*sqrt(b/a)*elliptic_f(arcsin(x*sqrt(b/a)), -1) - (2*b^2*x^ 
4 - 3*a^2)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a))/(a^2*x^7)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=\int \frac {\sqrt {a - b x^{2}} \sqrt {a + b x^{2}}}{x^{8}}\, dx \] Input:

integrate((-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2)/x**8,x)
 

Output:

Integral(sqrt(a - b*x**2)*sqrt(a + b*x**2)/x**8, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{x^{8}} \,d x } \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^8, x)
 

Giac [F]

\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{x^{8}} \,d x } \] Input:

integrate((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8,x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)/x^8, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {a-b\,x^2}}{x^8} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^8,x)
 

Output:

int(((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a-b x^2} \sqrt {a+b x^2}}{x^8} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}-2 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} x^{12}+a^{2} x^{8}}d x \right ) a^{2} x^{7}}{5 x^{7}} \] Input:

int((-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/x^8,x)
 

Output:

( - sqrt(a + b*x**2)*sqrt(a - b*x**2) - 2*int((sqrt(a + b*x**2)*sqrt(a - b 
*x**2))/(a**2*x**8 - b**2*x**12),x)*a**2*x**7)/(5*x**7)